The key difference is the $ \frac{1}{4\pi\epsilon_0} $, with $ \epsilon_0 $ in the SI formulation of charge, being vaccuum permittivity with units $ (charge)^2(time)^2 (mass)^{−1}(length)^{−3} $. This satisfies the unit cancellation, and in the SI system makes the electric constant $ \mu_0 $ and $ \epsilon_0 $ now derived units. (See Vacuum Permittivity or SI Unit Redefinition)

For example, Coulomb's law in Gaussian units appears simple:

where F is the repulsive force between two electrical charges, Q1 and Q2 are the two charges in question, and r is the distance separating them. If Q1 and Q2 are expressed in statC and r in cm, then F will come out expressed in dyne.
By contrast, the same law in SI units is:

where $ \epsilon_0 $ is the vacuum permitivity, a quantity with dimension, namely (charge)2 (time)2 (mass)−1 (length)−3. Without $ \epsilon_0 $ , the two sides could not have consistent dimensions in SI, and in fact the quantity $ \epsilon_0 $ does not even exist in Gaussian units. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law simply by the judicious choice of units. In SI, $ \frac{1}{\epsilon_0} $, converts or scales flux density, D, to electric field, E (the latter has dimension of force per charge), while in rationalized Gaussian units, flux density is the very same as electric field in free space, not just a scaled copy.
Since the unit of charge is built out of mechanical units (mass, length, time), the relation between mechanical units and electromagnetic phenomena is clearer in Gaussian units than in SI. In particular, in Gaussian units, the speed of light $c$ shows up directly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it only shows up implicitly via the relation .

- Wikipedia: Gaussian Units

Yes, I would argue that 'fundamental quantities' are indeed arbitrary, as are many of our choices, such as base-10 number systems. This is illustrated well on the Golden Record we put on voyager spacecraft, for decoding by other intelligent life; we show how fast to spin the record by relating time units in the fundamental transition of the hydrogen atom:

I'd then add that we have tried to make them as least-arbitrary (to us) as possible, but there's no reason that some other intellegence would have different 'fundamental unit' definitions and scalings, or whatever 'arbitrary' units they came up with. We could use $ (time)^{-1} $ or 'period' as our fundamental timing unit, and change all the other derived units to follow, if we wanted.

Perhaps you are confused between dimension and unit.

Note that $cm$ and $m$ are different units but have same dimension of length. See? It's simple. They have only different magnitudes.

You have to understand that you cannot subtract or add 1 kg from 1 metre. Makes no sense, right?

Suppose you want to know about speed. You know that it is $\frac {distance}{time}$ Hence its units are $\frac {m}{s}$ and its dimensions are clear by formula.

You see that if a formula says that $1 kg = 1 s$ , It makes no sense, right?

So you check what the thing you wanna find about depends on and let analyse how to multiply and divide them to get the same dimension of thing you are looking for.

Note that this still will not give you perfect formula as $\frac {distance}{time}$ and $2*\frac {distance}{time}$ have same dimensions, you will be short of a constant.

Also it cannot predict equations like $v=u+at$

## Best Answer

I think one should be careful with the "fundamental-ness" of quantities as regards measurement vs properties. What we call a base quantity does not

necessarilyequate nor connote the quantity with a fundamental property. You yourself note that temperature is a complicated concept when put under scrutiny.A physical quantity is something physical that can be measured in some way and that we then quantify. The choice of such units are arbitrary and tend to have a lot to do with history (although that's about to change). The list of four "properties" you have given are better phrased as base units and if you're talking base units, you have seven as given by the SI: metre, kilogramme, second, Ampere, Kelvin, mole and candela.

As units, they are a bit more "solid" to breaking down (and this allows us to perform useful things like dimensional analysis) but are by no means impervious; your system of base units can be anything you want, it merely has to be coherent and consistent. Of course, you'd have to convince the world to adopt your system as opposed to the one that's already in place to get anyone to work efficiently with you, but that's beside the point.

Charge is a fundamental property but not, I believe, in the way you've phrased it. The electromagnetic force is a fundamental property because it is one of the four fundamental forces: it appears irreducible to a more basic form or interaction. But it's

notbecause of the base unit of charge. It arises from discrete symmetries, is additive, countable and is relativistically invariant. In fact, it's the other way around, the unit of charge is (will) be because of the elementary electric charge.Thinking from the roots of units, what

ischarge ? "Amount of zap-zappiness", sure, but its unit is the Coulomb. What is a Coulomb ? Why, it's the amount of current transported by an Ampere in one second ! What's an Ampere ? An Ampere is flow of constant current such that, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length.Is then your charge a derivative of the metre and kilogramme ? Is then your candela merely a weighted summation of contributions of wavelengths of light ? Is then your kilogramme defining your newton, or should it be the other way around ? In special relativity, there's even a notion of expressing time in dimensions of distance, $ct$, the speed of light multiplied by time. Should we take that, then ?

Base units are base units, determined largely by consensus, convenience and usefulness. "Fundamentality" often comes from deeper arguments: symmetries, theories, equivalences, laws, etc. The "fundamentality" of a thing might inform the unit choice of something, but not vice versa.